Momentum

Cambridge IGCSE Physics 0625 · Extended

Syllabus points

• Define momentum and recall p = mv.
• Define impulse and recall that impulse equals the change in momentum.
• Apply the principle of conservation of momentum to collisions in one dimension.
• Define resultant force as the change in momentum per unit time.

Momentum

momentum = mass x velocity, so p = mv

Units are kg m/s. There is no special name for this unit, which is a small help: if your answer carries any other unit, the working is wrong.

Momentum is a vector. It inherits its direction from the velocity, and that is the single most important fact in this subtopic. Mass is a scalar; multiplying a vector by a scalar leaves a vector.

The equation is a product, not a quotient. An option reading p = m / v is not a physical quantity at all, and its units, kg divided by m/s, are nothing you will ever meet.

Why the direction matters

Take one direction as positive and hold it for the whole question. Anything moving the other way carries a negative momentum.

This is what allows momentum to cancel. Two objects can approach each other with equal and opposite momenta, collide, and both stop, without any momentum disappearing: the total was zero before and is zero after.

It is also why a bouncing ball is the standard question.

Worked example. A 0.50 kg ball hits the floor at 10 m/s and rebounds at 8.0 m/s. The collision lasts 0.50 s. Find the average force.

Take upwards as positive. Initial velocity = −10 m/s. Final velocity = +8.0 m/s. Change in velocity = 8.0 − (−10) = 18 m/s upwards. Change in momentum = 0.50 x 18 = 9.0 kg m/s upwards. Force = 9.0 / 0.50 = 18 N upwards.

Subtracting the speeds as though both were in the same direction gives 2.0 m/s and an answer nine times too small, and that answer is always on offer. When anything bounces the speeds add, because the object has to be stopped and then driven back the other way.

The force is upwards because the floor has to push the ball back.

Force as rate of change of momentum

resultant force = change in momentum / time taken

This is Newton's second law in its more general form, and F = ma is the special case where the mass is constant.

Rearranged, force x time = change in momentum. The product force x time is the impulse.

impulse = Ft = change in momentum = mv − mu

The practical consequence is the reason for crumple zones, airbags, crash mats and bending your knees when you land. The change in momentum is fixed by the collision, so increasing the time of the collision reduces the force. Nothing reduces the momentum change; you can only spread it out.

Conservation of momentum

In the absence of external forces, the total momentum before a collision equals the total momentum after.

total momentum before = total momentum after

This holds for collisions where the objects bounce apart and for those where they stick together.

Worked example. A 0.16 kg ball moving at 0.50 m/s strikes a stationary 0.10 kg ball. The second ball moves off at 0.50 m/s. Find the speed of the first ball afterwards.

Before: 0.16 x 0.50 + 0.10 x 0 = 0.080 kg m/s. After, the second ball carries 0.10 x 0.50 = 0.050 kg m/s. So the first ball carries 0.080 − 0.050 = 0.030 kg m/s. Its speed = 0.030 / 0.16 = 0.19 m/s.

Work in momentum throughout and convert back to a speed only at the very end.

The first ball stops dead only when the two masses are equal, which is a special case and not the general rule.

Momentum and kinetic energy

Momentum is conserved in every collision. Kinetic energy is not.

• In an elastic collision, kinetic energy is also conserved.
• In an inelastic collision, some kinetic energy becomes thermal energy, sound, or work done deforming the objects. Momentum is still conserved.

Most real collisions are inelastic. If a question mentions that the objects make a noise, or that they stick together, or that they are deformed, it is telling you kinetic energy is not conserved, and any option claiming it is can be struck out.

Newton's third law in this topic

The force each object exerts on the other is equal and opposite, and they act for the same length of time. So the impulses are equal and opposite too, and one object gains exactly the momentum the other loses.

That is why statements such as "the change in momentum of X equals the change in momentum of Y" are wrong as written. The changes are equal in size and opposite in direction, and dropping the minus sign changes the meaning.

Common mistakes

• Subtracting the speeds in a rebound instead of adding them.
• Forgetting that momentum is a vector and ignoring signs.
• Saying kinetic energy is conserved in a collision the question has told you is noisy or sticky.
• Writing the changes in momentum of two colliding objects as equal without the opposite sign.
• Assuming the incoming object always stops after a collision.
• Quoting momentum in newtons, or impulse in kg m/s² .
• Thinking a crumple zone reduces the change in momentum rather than the force.